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W-Algebras and Hamiltonian Equations

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W-Algebras and Hamiltonian Equations W-algebras and Hamiltonian equations Alberto De Sole Universit`adi Roma La Sapienza Dobbiaco, february 17-22, 2019 1 / 29 Lecture 1: Review of classical and quantum Hamiltonian mechanics 2 / 29 General Motivation: we will be mainly concerned with the role and applications of the concept of symmetry in mechanics, from a purely algebraic point of view. Recall that: I classical mechanics deals with the dynamics of particles, rigid bodies, etc, I classical field theory deals with the dynamics of continuous media (fluids, plasma), and fields such as the electromagnetic field, gravity, etc. It can be viewed as an infinite dimensional analogue of the classical mechanics. We shall deal with the Hamiltonian formulation of both theories. 3 / 29 1. Standard Hamiltonian equations. In classical mechanics, the state of a physical system is a point (q; p) in a 2n-dimensional space of states M (q = (q1; : : : ; qn) = positions, p = (p1; : : : ; pn) = momenta). The dynamics (=time evolution) of the physical system is described in terms of the Hamiltonian function H(q; p) (=energy of the system). The time evolution (q(t); p(t)), t > 0, is solution of the Hamilton equations: @H @H q_i = ; p_i = − : @pi @qi Example: Newton's law of dynamics F = ma defines a Hamilton evolution, where F = −∇U is the force field associated to an energy potential U(q), and the Hamiltonian jpj2 H = Ek + U is the sum of the kinetic energy Ek = 2m and the potential energy U(q) 4 / 29 The space of physical observables is the space C1(M) of all smooth functions of the space of states M. For example: the coordinates qi and pi, the energy H(q; p), are physical observables. C1(M) is a commutative associative algebra (with respect to the usual product of functions). It has the Poisson bracket n X @f @g @f @g ff; gg = − : @p @q @q @p i=1 i i i i The Hamiltonian equations can then be rewritten in terms of the Poisson bracket: dq dp i = fH; q g ; i = fH; p g dt i dt i By the chain rule of derivation, for f 2 C1(M): df = fH; fg dt 5 / 29 2. Hamiltonian evolution on a Poisson manifold The formulation of Hamilton equations in terms of the Poisson bracket allows us generalize the theory. The space of states of a classical mechanical system is a Poisson manifold: a smooth manifold M (NOT nec. even dim.) with a Poisson bracket {· ; ·} on C1(M), satisfying skewsymmetry ff; gg = −{g; fg the Jacobi identity ff; fg; hgg − fg; ff; hgg = fff; gg; hg and the Leibniz rule ff; ghg = ff; ggh + gff; hg (Equivalently: C1(M) is a Poisson algebra.) Exercise 0: the standard Poisson bracket satisfies these axioms The Hamilton equations associated to the Hamiltonian H are: df = fH; fg dt 6 / 29 Example: The dynamics of of a rigid body about its center of mass in absence of external forces is: (1) I2 − I3 I3 − I1 I1 − I2 Π_ 1 = Π2Π3 ; Π_ 2 = Π3Π1 ; Π_ 3 = Π1Π2 I2I3 I3I1 I1I2 Πi = IiΩi, i = 1; 2; 3, =angular momenta(Ω i = angolar velocities, Ii = moments of inerzia). It has the Hamiltonian form with Poisson bracket (2) ff; gg(Π) = Π · (rf × rg) and Hamiltonian function 1Π2 Π2 Π2 (3) H(Π) = 1 + 2 + 3 2 I1 I2 I3 Exercise 1: Check that (2) is a Poisson bracket, and that (1) are the Hamiltonian equations for H in (3). 7 / 29 Example: The Kirillov-Kostant Poisson structure on g∗ A generalization of the rigid body Poisson bracket is the Kirillov-Kostant bracket associated to a Lie algebra g. The underlying Poisson manifold is g∗. The Poisson bracket on C1(g∗) is (4) ff; gg(ζ) = hζj[dζ f; dζ g]i where h· j ·i is the pairing of g∗ and g, and the differential ∗ dζ f 2 g is defined by (ζ; ξ 2 g ): 1 lim (f(ζ + εξ) − f(ζ)) = hξjdζ fi "!0 " Exercise 2: Check that the Kirilov-Kostant Poisson bracket (4) satisfies skewsymmetry, Jacobi identity and the Leibniz rule. Exercise 3: Check that the Poisson bracket of the rigid body is a special case of the Kirillov-Kostant Poisson bracket when g = so3(R) 8 / 29 Remark: From a purely algebraic point of view, we can replace 1 ∗ ∗ smooth functions C (g ) by polynomial functions C[g ] ' S(g). This identification maps X P = coeff.a1 : : : as 2 S(g) to the polynomial function X ∗ P (ζ) = coeff.hζja1i ::: hζjasi ; ζ 2 g Exercise 4: Under this identification, the Kirillov-Kostant Poisson bracket (4) corresponds to the canonical Poisson bracket of the symmetric algebra S(g), obtained by extending the Lie algebra bracket of g to S(g) by the Leibniz rules. 9 / 29 3. Hamiltonian equations in classical field theory Classical field theory describes the evolution of continuous media, such as fluids, strings, electromagnetic field, etc. The state of the system is described by a field '(x), i.e. a function on x 2 S (=the space or space-time). Hence, the space of states M = Fun(S) is 1-dimensional. Assumptions: I For simplicity, we assume that S is one-dimensional. I '(x) is smooth and rapidly decreasing; hence all integrals are defined, and we can perform integration by parts. The physical observables are local functionals, F : M! R: Z F (') = f('(x);'0(x);:::;'(n)(x))dx S where the density function f depends on '(x);:::;'(n)(x) only. 10 / 29 To give a Hamiltonian formulation of classical field theory, we assume that the space of states M is a Poisson manifold. This means that we have a local Poisson bracket {· ; ·} on the space F(M) of local functionals (=physical observables). It is a Lie algebra bracket on F(M) of the form Z δG 0 (n) δF fF; Gg(') = K('(x);' (x);:::;' (x); @x) S δ'(x) δ'(x) 0 (n) d The Poisson structure K('(x);' (x);:::;' (x); dx ) is a finite order differential operator. (@x = total derivative w.r.t. x). δF The variational derivative δ'(x) is defined by 1 Z δF lim (F (' + ψ) − F (')) = (x)dx !0 S δ'(x) for every test function (x) 2 M. 11 / 29 Exercise 5: Explicit formula for the variational derivative: δF X @f = (−@ )n δ'(x) x @'(n)(x) n2Z+ Example: The Gardner-Faddeev-Zakharov(GFZ) local Poisson-bracket is given by Z δG δF (5) fF; Gg(') = @x S δ'(x) δ'(x) i.e., it has Poisson structure K = @x. Exercise 6: Check that the GFZ local Poisson bracket (5) is a Lie algebra bracket on F(M). 12 / 29 The dynamics (=time evolution) of the physical system is described in terms of the Hamiltonian functional H(') 2 F(M), describing the energy of the system in the state '(x). It is given by the Hamiltonian equations dF = fH; F g dt where F (') is a local functional of '(x). We can also write an equation for the evolution of the coordinate variable '(x; t). By the definition of variational derivative, we have dF Z δF d = '(x; t) ; dt S δ'(x) dt Combining this with the form of the local Poisson bracket, we get d δH '(x; t) = K('(x);'0(x);:::;'(n)(x); @ ) dt x δ'(x) 13 / 29 Example: The famous Korteveg-de Vries (KdV, 1985) equation @' @' @3' (6) = 3'(x; t) + c @t @x @x3 describes the evolution of waves in shallow water. It is the Hamiltonian equation associated to the GFZ Poisson bracket on M and the Hamiltonian functional Z 1 c H(') = '(x)3 − '0(x)2 dx S 2 2 Exercise 7: Check the above statement. 14 / 29 4. Quantum mechanics In quantum mechanics the space of states of a system is a Hilbert space, i.e. a complex vector space V with a positive definite hermitian inner product h· j ·i. A state is a normalized vector jvi2 V , such that hvjvi = 1. The physical observables are operators A 2 End(V ), which are selfadjoint: A = Ay. The physical meaning is as follows: if the state of the system is an eigenvector jvi2 V of A of eigenvalue λ 2 R, then the result of the measurement of the observable A is λ. By the Spectral Theorem, any state jvi is linear combination of P orthonormal eigenvectors with real eigenvalues: jvi = i cijvii, with Ajvii = λijvii. Then, the result of a measurement of the 2 observable A is λi with probability jcij . 15 / 29 The Hamiltonian operator H describes the energy of the system and defines its dynamics. There are two alternative ways to introduce the dynamics: I In the Schroedinger picture, the observables do not evolve, while the states evolves by the Schroedinger equation d (7) j i = Hj i dt t t I In the Heisenberg picture, the the state does not evolve, while the observables evolve by to the Heisenberg equation: d (8) A(t) = [H; A(t)] dt where [H; A] = HA − AH is the commutator. Note: This is the \quantum version" of the Hamiltonian equation of classical mechanics. Exercise 8: Show that the Schroedinger equation (7) and the Heisenberg equation (8) are equivalent in the following sense: the evolution of the matrix elements h'jAj i is the same. 16 / 29 5. Algebraic structures of classical and quantum mechanics From a purely algebraic point of view, when considering a classical mechanic physical system, we ignore completely the underlying configuration space M, and we just retain the algebraic structure of the space of functions C1(M).
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